Problem: Simplify. Multiply and remove all perfect squares from inside the square roots. Assume $a$ is positive. $3\sqrt{5a}\cdot 8\sqrt{35a^2}=$
Let's start by multiplying the factors within and without the square roots: $\begin{aligned} 3\sqrt{5a}\cdot 8\sqrt{35a^2} &=3\cdot 8\cdot\sqrt{5a}\cdot\sqrt{35a^2} \\\\ &=24\sqrt{175a^3} \end{aligned}$ Now we remove all perfect squares from inside the square root: $\begin{aligned} 24\sqrt{175a^3}&=24\sqrt{5^2\cdot a^2\cdot 7a} \\\\ &=24\sqrt{5^2}\cdot\sqrt{a^2}\cdot\sqrt{7a} \\\\ &=24\cdot 5\cdot a\sqrt{7a} \\\\ &=120a\sqrt{7a} \end{aligned}$ In conclusion, $3\sqrt{5a}\cdot 8\sqrt{35a^2}=120a\sqrt{7a}$